Integrated rate laws Knowledge-Based Study

Study notes

1. Connecting Concentration and Time

   Differential rate laws describe how the rate of a chemical reaction depends on the instantaneous concentrations of reactants, such as $\text{Rate} = k[\text{A}]^n$. However, in experimental laboratory work, measuring instantaneous reaction rates directly can be challenging.

   Integrated rate laws are derived by integrating differential rate laws with respect to time. They provide a direct mathematical relationship between the reactant concentration and elapsed time ($t$). This allows chemists to:

   • Determine the concentration of a reactant remaining at any given time.
   • Calculate the time required for a reaction to proceed to a specific extent.
   • Identify the reaction order experimentally by graphical analysis of concentration-versus-time data.

   Zero-Order Kinetics

   For a reaction $\text{A} \rightarrow \text{products}$, a zero-order process means the reaction rate is completely independent of the concentration of the reactant:

   $$-\frac{d[\text{A}]}{dt} = k$$

   Integrated Rate Law

   Integrating from time $t = 0$ (where $[\text{A}] = [\text{A}]_0$) to time $t$ gives:

   $$[\text{A}]_t = -kt + [\text{A}]_0$$

   Key Features

   • Linear Plot: A plot of $[\text{A}]_t$ versus $t$ yields a straight line with a slope equal to $-k$ and a y-intercept of $[\text{A}]_0$.
   • Units of Rate Constant ($k$): $\text{M}\cdot\text{s}^{-1}$ (or $\text{mol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$).
   • Half-Life ($t_{1/2}$): Setting $[\text{A}]_t = \frac{[\text{A}]_0}{2}$ gives $$t_{1/2} = \frac{[\text{A}]_0}{2k}$$ Notice that for zero-order reactions, the half-life is directly proportional to the initial reactant concentration.

   First-Order Kinetics

   In a first-order reaction, the rate is directly proportional to the concentration of a single reactant raised to the first power:

   $$-\frac{d[\text{A}]}{dt} = k[\text{A}]$$

   Integrated Rate Law

   Integrating this differential equation yields the logarithmic relationship:

   $$\ln[\text{A}]_t = -kt + \ln[\text{A}]_0$$

   This can also be expressed in exponential form:

   $$[\text{A}]_t = [\text{A}]_0 e^{-kt}$$

   Key Features

   • Linear Plot: A plot of $\ln[\text{A}]_t$ versus $t$ produces a straight line with a slope of $-k$ and a y-intercept of $\ln[\text{A}]_0$.
   • Units of Rate Constant ($k$): $\text{s}^{-1}$.
   • Half-Life ($t_{1/2}$): Setting $[\text{A}]_t = \frac{[\text{A}]_0}{2}$ leads to:$$\ln\left(\frac{1}{2}\right) = -kt_{1/2} \implies t_{1/2} = \frac{\ln(2)}{k} \approx \frac{0.693}{k}$$
   • Crucial Observation: The half-life of a first-order reaction is completely independent of the initial concentration $[\text{A}]_0$. Each successive half-life takes the exact same amount of time.

   Second-Order Kinetics

   For a reaction that is second order in reactant $\text{A}$, the rate law is:

   $$-\frac{d[\text{A}]}{dt} = k[\text{A}]^2$$

   Integrated Rate Law

   Integrating this expression yields:

   $$\frac{1}{[\text{A}]_t} = kt + \frac{1}{[\text{A}]_0}$$

   Key Features

   • Linear Plot: A plot of $\frac{1}{[\text{A}]_t}$ versus $t$ yields a straight line with a positive slope of $+k$ and a y-intercept of $\frac{1}{[\text{A}]_0}$.
   • Units of Rate Constant ($k$): $\text{M}^{-1}\cdot\text{s}^{-1}$ (or $\text{L}\cdot\text{mol}^{-1}\cdot\text{s}^{-1}$).
   • Half-Life ($t_{1/2}$): Setting $[\text{A}]_t = \frac{[\text{A}]_0}{2}$ gives:$$t_{1/2} = \frac{1}{k[\text{A}]_0}$$
   • Crucial Observation: The half-life is inversely proportional to the initial concentration. As the reaction proceeds and reactant is consumed, each successive half-life doubles in duration compared to the preceding one.

   Determining Reaction Order from Kinetic Data

   To identify the order of an unknown reaction experimentally, concentration data measured over time are tested using three distinct plots:

   1. Test for Zero Order: Plot $[\text{A}]$ vs. $t$. If linear, order = 0, and $\text{slope} = -k$.
   2. Test for First Order: Plot $\ln[\text{A}]$ vs. $t$. If linear, order = 1, and $\text{slope} = -k$.
   3. Test for Second Order: Plot $\frac{1}{[\text{A}]}$ vs. $t$. If linear, order = 2, and $\text{slope} = +k$.

   Comparison of Integrated Rate Laws

   • Zero Order: $[\text{A}]_t = -kt + [\text{A}]_0$ | Linear plot: $[\text{A}]$ vs. $t$ | Half-life: $t_{1/2} = \frac{[\text{A}]_0}{2k}$ | Units of $k$: $\text{M}\cdot\text{s}^{-1}$
   • First Order: $\ln[\text{A}]_t = -kt + \ln[\text{A}]_0$ | Linear plot: $\ln[\text{A}]$ vs. $t$ | Half-life: $t_{1/2} = \frac{0.693}{k}$ | Units of $k$: $\text{s}^{-1}$
   • Second Order: $\frac{1}{[\text{A}]_t} = kt + \frac{1}{[\text{A}]_0}$ | Linear plot: $\frac{1}{[\text{A}]}$ vs. $t$ | Half-life: $t_{1/2} = \frac{1}{k[\text{A}]_0}$ | Units of $k$: $\text{M}^{-1}\cdot\text{s}^{-1}$